The subject line gives away my ulterior motive in the original posting.
But first, I was astonished at getting 11 personal e-mail messages about my
Barber story, before I received the issue of Linguist that contained
it. I did intend to start something, but did not expect that much of a
reaction. The count now stands at 15, plus those who wrote their
thoughts to Linguist.
Second, I had been zealously searching for the book where I read the
Spanish Barber paradox. I needed the precise wording of the paradox,
for if a linguistic illusion there was, it must have depended on the
wording. In vain. I thought I had it when, in the index of Lewis
Carroll's "Symbolic Logic" I saw "Barber-Shop Paradox". But it wasn't.
Eventually, that is, yesterday (28/9) I found it mentioned in a slim
Penguin book, "Vicious Circles and Infinity", by Patrick Hughes and
George Brecht, but the discussion I remembered was missing. I suspect
it might have been in a book by Bertrand Russell, the title of which I
cannot remember, where he told the story of Hegel publishing a proof
that there could be only seven planets shortly before Uranus was
discovered, and how poor Hegel scoured the bookshops, frantically
buying back copies of his monograph [Note 1].
The Spanish Barber paradox, some correspondents pointed out (that
refreshed my memory), was meant to do with the "catalog of catalogs
that do not list themselves." Try to write a catalog of the catalogs
that do not list themselves and, just before you go to the printer with
your manuscript in your hot little hand, ask yourselves "Ah, but should
I have listed my catalog itself in there?". A clearly stated paradox.
"Vicious Circles and Infinity" mentions, p.12, how Russell proposed to
dispose of the contradiction:
"it seemed to me that a class sometimes is, and sometimes
is not, a member of itself. The class of teaspoons, for
example, is not another teaspoon, but the class of thing
that are not teaspoons, is one of the things that are
not teaspoons."
I certainly did not see what the Spanish Barber had to do with classes
and sets when I first read it.
Consider the wording of the paradox as taken from "Vicious Circles",
which is probably what Bertrand Russell wrote verbatim:
A man of Seville is shaved by the Barber of Seville if
and only if the man does not shave himself? Does the
Barber of Seville shave himself?
Allow me to rephrase it, less sloppily ("a man of Seville" does not
mean "every man of Seville"; the Barber of Seville need not be a man of
Seville, he could be of Malaga; "if and only if" does not clearly
convey that he must be clean-shaven)
Every man in Seville will be shaved today by one certain
man in Seville unless he shaves himself. Will that
certain man in Seville shave himself today?
Does that sentence express anything other than the original? I think
not. But that certain man will shave himself today and all the others
who don't and the paradox has vanished. Where did it go? Where was it
hiding in the first place?
Let me turn Seville into an Indian village (the caste system will
help). We have a caste of men, the Gillettes, who shave themselves.
Another, the Schicks, think it below their dignity, and have themselves
shaved by a third caste, obviously the dregs of the dregs, called...
the Barbers. The Gillettes are not allowed, of course, to get shaved, a
privilege reserved for the superior Schicks. What of the Barbers? This
scum of the earth is not allowed to have itself shaved by other Barbers
(Now who do you think you are, Barber? One of us Schicks? Off with your
head!) They are not allowed to shave themselves either, that is the
Gillettes' prerogative. Those rules take care of the "if and only if"
bits in the Spanish Barber paradox (or exclusive or's, if you prefer to
word it so). In good logic you would conclude that those Barbers all
wear long beards or, as a few have jocularly said, must be women. But
the Spanish Barber's paradox, because it claims to be a paradox,
whatever its exact wording, rules that out: male and clean-shaven they
must be, and using tweezers, smouldering walnut shells, and other
untonsorial artifices on themselves is out too. An impossible state of
affairs.
Now what is this? It is me asking you to solve these equations:
y = 2x
x = y-1
and, just as you are about to say "ha, ha! x= 1 and y = 2", adding:
no, because you should also satisfy this condition
that x = y.
That is precisely, I think, what Stavros Macrakis writes in Linguist:
"the original definition is flawed". There is no paradox, no mystery,
just an error. What is it, then, that endowed it with such mystery?
The first time I read the Barber paradox, I remember saying to myself
immediately "this is stupid, it would have me believe that the barber
is not himself". How does that lead to the catalog of catalogs that do
not list themselves?
Turn "Indian village" into "library", "shave" into "refer to", "man"
into "book", "caste" into "floor".
On the "Gillette" floor are stored books that refer to themselves and
no other; on the "Schick" floor books that refer to none, but are
referred to by others; on the "Barber" floor books that refer to those
on the "Schick" floor. So there is the analogy: one day the librarian
writes a book that refers to all the books on the Barber floor so that
none be unreferenced; one day the rajah appoints a man to shave all the
barbers so that none be hirsute. The next question is: has such a book
its place on any of the existing floors, has such a barber a status in
any of the existing castes? A bit of reflection shows that no, the
rules have to be changed. The barber of barbers must belong to a new
caste, the catalog of catalogs that do not list themselves hide on a
different floor. A boxful of Bertrand Russell teaspoons is not a
teaspoon.
My initial question still remains unaswered: what made it so difficult
to recognize?
Bacchus inspiring, think of a wine bottle that would contain the labels
of all the wine bottles that do not contain their own label. Let us
look at one of those bottles. It is a wine bottle, with wine labels
stuffed into it. Is its own label inside? But wait, what do we mean by
"own label"? The labels inside symbolize wines, the label on this
bottle itself symbolizes a bottleful of labels. To equate the label on
the bottle with any label inside it we must ignore what those labels
refer to and treat them as nothing but pieces of paper with doodles on
them. That is why I mentioned a linguistic illusion: a paradox arises
only if we take sign and referent (signifiant and signifie') to be one
and the same. Your photograph becomes you and if I tear it up you die.
But in so doing we have destroyed the very notion and mechanism of
reference. We should not be allowed to lay claim to logical thinking
any more, only magical. Am I setting up a straw man for my foul
purposes?
Consider chapter 16 of Hofstadter's "Goedel, Escher, Bach: an Eternal
Golden Raid". Hofstadter opens with five sentences to introduce us to
the intricacies of self-referencing:
(1) This sentence contains five words.
(2) This sentence is meaningless because it is
self-referential.
(3) This sentence no verb.
(4) This sentence is false.
(5) The sentence I am now writing is the sentence you are
now reading.
He discusses only sentence (4), of which he writes:
The difficulty is perhaps underlined when a sentence
such as number 4 is presented to someone naive about
paradoxes, and linguistic tricks, such as a child. They
may say, "What sentence is false?" and it may take a bit
of persistence to get across the idea that the sentence
is talking about itself.
"This sentence is false (true)" is equivalent to "It is false (true)
that <some statement>" with the noun phrase "this sentence" referring
to <some statement>. But, as someone naive about paradoxes and
linguistic tricks would rightly remark, "What statement?". There is no
statement there. Even granting that "the sentence is talking about
itself", all it says is "It is false that it is false that is false
that..." about, ultimately, nothing.
Take sentence (3) now:
This sentence no verb.
"This sentence no verb" is not a sentence, but what is left after you
have removed the verb from a sentence. What happens? As often when
confronted with a misspelt word we try to figure out what it was meant
to be and here, we easily reconstruct: "This sentence has no verb",
referring to, and separate from, the ill-formed utterance "This
sentence no verb". No self-referencing there. The mechanism by which we
interpret that particular utterance, and are conned into agreeing with
it, is nowhere discussed. It reminds me very much of the forced-card
trick of conjurers. We are made to pick a word, "has" or "contains",
putting it back into the deck, and voila! "This sentence has no verb".
Ah, but what if you had been aware of the trick and picked the wrong
word on purpose: "This sentence lacks no verb"?
Now take sentence (1):
This sentence contains five words.
That sentence does indeed consist of five words, and this should give
us an inkling of the sleight of hand that is being performed.
What has been happening? We are being led into accepting as "sentence"
a string of words separated by spaces and terminated by a period. For
instance:
This sentence no verb.
At the same time we are made to reflect on the form of some strings of
words and on their meanings, or what their meanings would be when, by
pruning here and grafting there, they have been made into well-formed
sentences. Form and content, sign and referent, signifiant and
signifie' have been inextricably confused. All along, our attention is
kept away from the fact that there are two sides to any utterance, its
form and its content and we are made to marvel at how sometimes form
matches content and sometimes not. Likewise a conjurer performs his
tricks by distracting his audience, but he does so consciously and does
not himself believe that he has proven the spontaneous generation of
rabbits out of top hats.
That error does not seem to be Hofstadter's alone.
"Vicious Circles" also opens with self-reference:
There are many propositions with are self-referential.
For instance:
This sentence has five words.
This is a sentence.
This sentence is written in the English language.
(Vicious Circles and Infinity, p.1)
Those propositions, however, refer to their printed expression in the
English language. If you say that they refer to themselves, you imply
that they are nothing but their printed form. [Note 2]
And now back to the Spanish Barber, driven to the bottle.
To turn the Spanish Barber into a paradoxical catalog of catalogs we
need only imagine that he keeps a list of his customers, a collection
of visiting cards in a bottle for example [Note 3]. Next we stick his
own visiting card on the bottle and ask "Does the bottle contain that
card?". Only then does the problem become that of the catalog of
catalogs that do not list themselves [Note 4] and a paradox is allowed
to arise as long as we insist on confusing sign and referent.
-------------------------
[Footnote 1] Hegel was born in 1770, Uranus discovered in 1781. Was
Bertrand Russell taking the mickey out of his readers, or is my memory
playing tricks?
[Footnote 2] "Vicious Circles" is replete with what I see as nothing
but hocus pocus and obfuscation of the same ilk. For instance:
Valdis Augstkalns has proposed an absolute refutation of
Jourdain's paradox. He suggests 'a strip of paper with
"The statement on the other side of this paper is true"
written on one side of it, and "The statement on the
other side if this paper is false" written on the other
side of it... One takes the paper, gives it a half
twist, and joins the ends to form a Moebius strip. The
serious and philosophically legitimate question is
transformed to 'Eminent members of the panel, which is
the other side of the paper?'
To dare call the question "serious and philosophically legitimate"
makes my blood boils. There is nothing to refute for there is nothing
there, but a mindless "It is true that it is false that it is true that
it is false that..." ad infinitum, without a proposition ever
concluding this parrot's babble. I was quite surprised to find that
book on my shelves: I was persuaded that I had thrown it out in
disgust.
[Footnote 3] He could keep them in his wallet, or their names in his
memory; he could dye their beards purple instead of shaving them, we
would still be back to the non-paradox of the bottle of labels.
[Footnote 4] How misleading the very wording is! A catalog does not
list itself, other catalogs, garden tools, nor thingagummy widgets, it
lists names of catalogs, garden tools, or thingagummy widgets.